In General Relativity it is energy that creates gravity, not just mass (rest energy). So yes forming a black hole of only photons is a theoretical possiblity.
I think you are correct that photons alone could not commence formation of a black hole in the current universe in naturally occuring conditions.
A non gravitational mechanism of achieving the energy density to create an event horizon would need to be employed. An immense array of lasers focused on single point would have to fire simultaineously.
Would a black hole thus formed be comprised of photons? No.
It has to be considered that beyond the event horizon, and approaching a hypothesised singularity, the original form this energy came in becomes irrelevant, matter will adopt the approriate form according to its "temperature" due to proximity to the singularity.
So we would need to place the energy (focus the lasers) to form a sphere the right radius from the singularity to have an energy density that would allow it to all remain photonic (r could be calculated I think).
(Noting that observations beyond the event horizon are in current theory impossible, what would be the point - behaviour would be indistinguishable from any other black hole to external observers).
Once an event horizon has formed, from the perspective of the external observer, a black hole grows to envelop more matter, rather than absorbing matter. (Time stops at the event horizon.) This matter/energy, then, probably (no observations made) maintains the density it had at the event horizon and may not be photonic. So, even if it was originally formed using photons, and by some mechanism the matter content remained photonic, if non photonic matter was enveloped the purity of the black hole would be permentantly disrupted.